Find the intervals in which the following functions are increasin… - Vidyadip

Question

Find the intervals in which the following functions are increasing or decreasing.
f(x) = 2x³ - 24x + 107

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Answer

f(x) = 2x³ - 24x + 107
f'(x) = 6x² - 24
= 6(x² - 4)
= 6(x + 2)(x - 2)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ 6(x + 2)(x - 2) > 0
⇒ (x + 2)(x - 2) > 0
[Since, 6 > 0, 6(x + 2)(x - 2) > 0 ⇒ (x + 2)(x - 2) > 0]
⇒ x < -2 or x > 2
$\Rightarrow\text{x}\in(-\infty,-2)\cup(2,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,-2)\cup(2,\infty).$
For f(x) to be decreasing, we must have,
f'(x) < 0
⇒ 6(x + 2)(x - 2) < 0
⇒ (x + 2)(x - 2) < 0
[Since, 6 > 0, 6(x + 2)(x - 2) < 0 ⇒ (x + 2)(x - 2) < 0]
⇒ -2 < x < 2
$\Rightarrow\text{x}\in(-2,2)$
So, f(x) is decreasing on $\text{x}\in(-2,2).$

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